By fabricating devices with micro- and nanoscale channels, we can confine molecules such as DNA
in a controlled fashion. This is of critical importance because this enables observation of individual
molecules (rather than groups of molecules) with well defined boundary conditions. These detailed
measurements thus complement measurements on ensembles of molecules made using more conventional
techniques.

The models described above for DNA in a bulk solution work well as long as the size of the channel is large as
compared to the radius of gyration of the molecule. When the channel size is smaller than the radius of gyration of
the molecule, or when the ends of the DNA are controlled by attaching the ends of the DNA molecule to a wall or a
particle whose location is controlled, the conformation of the molecule is affected by these external forces.
Figure 14.10 shows DNA confined in a nanochannel, with the attendant change in the polymer configuration caused
by the confining geometry.

The importance of DNA dynamics in confining geometries is threefold: first, geometries that are small as
compared to the molecule radius of gyration offer potential improvements in our ability to manipulate,
sort, and separate DNA; second, studying a well defined and well known polymer in these geometries
enables fundamental study of polymer physics; and finally, DNA is often in a confining geometry in
living cells, and these geometries may have important implications for DNA’s biological function.

Figure 14.10: DNA confined in a nanochannel.

14.5.1 Energy and entropy of controlled polymer extension

The ends of DNA molecules can be attached to surfaces or microbeads, through which the end-to-end length can be
controlled using external forces. Such an experiment highlights DNA’sentropic spring constant. This spring constant
is not equal to the spring constant of the individual molecular bonds, nor the spring constant used in a bead-spring
model. Rather, this spring constant illustrates the entropic forces on the DNA molecule as a whole. Figure 14.11
shows a DNA molecule whose ends are controlled by attachment to a wall and manipulation of a microbead.

Figure 14.11: A DNA molecule whose ends are controlled by attachment to a wall and manipulation of a
microbead.

Consider the Gaussian bead-spring chain. From the distribution of ℓe, we can show using the Boltzmann
entropy relation (see Exercise 14.20) that the instantaneous entropyof the DNA molecule is given
by

(14.50)

where S is the instantaneous entropy of the molecule. This relation shows how entropy is related to the end-to-end
length of the molecule. If we pulled a DNA molecule tight, so that ℓe=ℓc, then ℓe2 would be much greater than
ℓcℓK, and the second (negative) term would be large in magnitude, making the entropy low. A DNA molecule pulled
tight has only one degree of freedom, i.e., its alignment with respect to the coordinate axes, and thus it has low
entropy. In contrast, a system with ℓe= 0 has enormous freedom, and thus high entropy. The high entropy state is
also the most likely.

We can cast the entropy result in terms of force by calculating the conformational free energy and equating the
force to the derivative of the freeenergy. First we write the conformational Helmholtz free energy, using
ΔA = ΔU -T ΔS:

(14.51)

where the molecule internal energy U isindependent of the conformation. Then we take the derivative of
Equation 14.51 with respect to ℓe to get the instantaneous restoring force F felt on the ends of the
molecule:

Note that the entropic force tends to pull the ends of the molecule together. For an ideal polymer, this
conformational spring constant is strictly entropic, and does not consider any electrostatic (i.e., energetic)
interactions between parts of the molecule.

From this, we can treat the ends of the molecule as if they were connected by a spring with a restoring force
F = -kℓe defined by a spring constantk, which has a value

Note that ℓK itself is a function of temperature and solution conditions. Equation 14.53 can also be written in
general for any contour distance Δs as k = 3kBT ∕ΔsℓK, indicating that entropy tends to act as a spring pulling the
components of the molecule together. This balances thermal fluctuations and self-avoidance, which tend to disperse
the parts of the polymer.

One end of an ℓc= 250 μm DNA molecule is tethered to the wall of a microdevice and the other end is tethered
to a 5 μm diameter polystyrene bead with density ρp= 1500kg∕m3. The system is at 300 K. A focused laser
beam (an optical tweezer) is used to pull the polystyrene bead to a distance of 200 μm from the wall. The laser is
turned off and the system returns to equilibrium. Modeling the polymer as a Gaussian bead-spring system with Kuhn
length ℓK= 100nm and ignoring Brownian motion of the bead, write the equations of motion for the
bead.

Solution:
Defining x as the distance from the wall, the entropic spring force for the DNA molecule is given
by

(14.54)

which, for the specified values, gives

(14.55)

There is a hydrodynamic drag on the polymer, but it is safely assumed smaller than that on the particle, especially
when the polymer is so fully extended. Thus the key hydrodynamic force is the Stokes drag on the particle, which
can be written using the bulk Stokes flow relation since the distance from the wall is more than ten times the
radius:

(14.56)

Note that the negative sign is used since u is the velocity of the particle and this relation is the drag force on theparticle caused by the fluid which, for water and a 2.5 μm radius particle, gives

(14.57)

The acceleration of the bead is given by the summed force divided by the particle mass, which is

(14.58)

With x and dx∕dt written in SI units, the equation of motion for the bead is thus

Note that the expression for the spring constant is valid for a Gaussian bead-spring chain of ℓc= 250 μm, but
this same expression would not be valid for a freely jointed chain.

These equations of motion predict that the bead will accelerate to a velocity near -2 μm∕s and move toward the
wall. As the particle moves toward the wall, the velocity will decrease proportional to the distance. As the particle
gets close to the wall, the Stokes relation will break down. In the physical system, Brownian motion would also add
random motion to the deterministic action of the polymer.

14.5.2 Energy and entropy of confinement for ideal polymers

The bead-spring molecule can describe the effects when a DNA molecule is confined. This is of particular interest
when considering the motion of DNA in nanochannels, which can be small as compared to . We
might confine DNA in nanochannels for a variety of reasons, including (1) validating DNA polymer
dynamic models by observing DNA behavior in a controlled environment with well-known geometry; (2)
physically separating DNA molecules based on contour-length dependent transport variations observed
at this length scale; or (3) extending the DNA molecule to sequence it or otherwise characterize it
chemically. We do not derive these relations directly, but we present the results. The properties of a
bead-spring chain that is confined in an idealized rectangular box with dimensions Lx, Ly, and Lz, and follow
from the change in the partition function and therefore entropy that a bead-spring chain has when
confined.

For a bead-spring molecule confined in this box, the partition function in a dimension i (where i = x,y,z) is given
by

(14.60)

where is the mean radius of gyration observed for the polymer in the bulk. The partition function of the system
is given by

(14.61)

and the total entropy is given by

(14.62)

We evaluate the partition function (and therefore entropy) in two limits. First, for the case where ≪ Li, i.e., the
case where the polymer is not confined by its surroundings, we get Zi= Li (see Exercise 14.9). For the case where
≫ Li, i.e., the case where the polymer is greatly confined, we get (see Exercise 14.10)

(14.63)

From this, we can show that the difference ΔS in entropy caused by finite effects in one dimension is negative
and given by

the change ΔA in Helmholtz free energy caused by finite effects is positive and given by

It has become the norm in the polymer physics literature to refer to systems with only one degree of confinement
(i.e., channels with nanoscale depth but large width) as nanoslits, whilesystems with two degrees of confinement
(i.e., nanoscale depth and width) are referred toas nanochannels. Many other communities do not make this
distinction.

In nanofluidic systems, we might typically confine DNA in a long, narrow nanochannel. If we assume that the
channel length is much larger than but the channel cross-section is square with width and depth d, the DNA is
confined in two dimensions and the free energy change is

Thus, the force required to confine DNA in the long narrow channel can be determined from free energy
considerations. This force causes DNA to tend to pool in large reservoirs rather than remaining in a
confined area, and is central to a number of separation techniques, e.g.,[150], which are discussed in
Chapter 15.

The relations above focus strictly on idealized bead-spring polymer chains. The work presented here can be
extended to address self-avoidance and volume exclusion, with improved accuracy.

14.5.3 DNA diffusion in confined geometries

Confinement changes the configuration of DNA, as described above, leading to changes also in DNA’s transport
properties. Most notably, DNA confined to geometries smaller than the radius of gyration is no longer
hydrodynamically coupled in a roughly spherical ball of diameter —rather, the molecule is extended into a
nonspherical conformation. In this case, the viscous mobility of this extended configuration is inversely proportional
to ℓc. In this case, the diffusivity of the DNA is inversely proportional to the contour length (D ∝ℓc-1) [142, 141].
The dependence of the diffusivity on the height h of a nanochannel is currently under dispute—different theories
(e.g., blob models, reflecting rod models) give different results. Experiments observe dependences ranging from
D ∝ h(1∕2) to D ∝ h23.

14.5.4 DNA electrophoretic mobility in confined geometries

Because DNA is free-draining during electrophoresis, the role of confining geometry on DNA electrophoretic
mobility is related to electrical double layer overlap, and can be described by similar equations.